fall down:To fail to meet expectations；lag in performance（失败：未能达到预想；在成就上落后）
例如：That young boy fell down on the job.那小伙子工作未能达到预想目标.
fall into:介入,开始.
usually you have just met,you start having a conversation or discussion with them（素不相识,一见面就能进入聊天或讨论某个问题）
例如：Over breakfast at my motel,I fell into conversation with the owner of a hardware shop.在 motel 用早餐时,我就同五金商店的店主聊天了.
还有‘分成,属于’含意.
例如：The book falls into three parts.这本书分三部分.